Subgroups — #31012782

Order 17152.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572641	C₂	1	1	yes	maximal, characteristic, complemented
2	#22571968	C₂	1	1	yes	maximal, characteristic
3	#22562433	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562403	C₂	1	2	yes	maximal, complemented
5	#22562351	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562308	C₂	1	2	yes	maximal, complemented
7	#22562249	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562148	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#22561915	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524066	C₂²	1	2	yes	—
13	#16864119	—	2	1	no	—
14	#16864068	—	2	1	no	—
15	C₂×#14523986	C₂²	1	2	yes	—
16	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
17	#16863801	C₂²	1	1	yes	characteristic
18	#16863773	C₂²	1	1	yes	characteristic
19	#16863736	—	2	2	no	—
20	#16863681	—	2	1	no	—
21	#16863634	C₂²	1	2	yes	—
22	#16863633	C₂²	1	2	yes	—
23	#16863596	—	2	1	no	—
24	#16863544	—	2	2	no	—
25	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
26	#16863433	C₂²	1	2	yes	—
27	#16863223	C₂²	1	2	yes	—
28	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
29	#16863200	C₂²	1	2	yes	—
30	#16863198	C₂²	1	2	yes	—
31	#16863198	C₂²	1	2	yes	—
32	#16863196	C₂²	1	2	yes	—
33	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	—
34	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
37	#366×C₆₇	C₂²	1	1	yes	characteristic
38	#358×C₆₇	C₂²	1	1	yes	characteristic
39	#358×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#11761629	—	2	2	no	—
42	#14524066	—	2	4	no	—
43	#14524066	—	4	2	no	—
44	C₂×#11761615	—	2	2	no	—
45	C₂×D₅₃₆	—	2	1	no	—
46	C₂×D₅₃₆	C₂³	1	2	yes	—
47	C₂×C₈×D₆₇	C₂³	1	2	yes	—
48	#14524043	—	2	1	no	—
49	#14524040	—	2	1	no	—
50	#14524040	—	4	1	no	—
51	#14524038	—	4	2	no	—
52	#14524031	—	4	1	no	—
53	#14524019	—	2	2	no	—
54	#14524018	—	2	2	no	—
55	#14524018	—	4	2	no	—
56	C₄×Dic₁₃₄	—	4	1	no	—
57	C₄×D₂₆₈	—	4	1	no	—
58	C₄×D₂₆₈	D₄	1	1	yes	characteristic
59	#14524001	—	2	2	no	—
60	#14523986	—	4	2	no	—
61	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
62	C₁₆×D₆₇	—	4	2	no	—
63	#14523977	—	2	1	no	—
64	#14523977	C₂³	1	2	yes	—
65	#14523974	—	2	2	no	—
66	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
67	#14523967	—	2	2	no	—
68	#14523963	—	2	2	no	—
69	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
70	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₆₇	—	2	1	no	—
73	C₄×Q₈×C₆₇	—	2	1	no	—
74	Q₃₂×C₆₇	—	2	2	no	—
75	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
77	#107×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	C₂³	1	1	yes	characteristic
79	#103×C₆₇	—	2	1	no	—
80	#103×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	#11761629	—	4	4	no	—
83	#11761629	—	8	2	no	—
84	C₂×Dic₁₃₄	—	4	1	no	—
85	C₂×D₂₆₈	—	2	2	no	—
86	C₂×D₂₆₈	—	4	2	no	—
87	C₂×D₂₆₈	—	4	1	no	—
88	C₂×D₂₆₈	—	4	2	no	—
89	C₂×D₂₆₈	—	4	1	no	—
90	C₂×D₂₆₈	D₈	1	1	yes	characteristic
91	C₂×C₄×D₆₇	—	4	2	no	—
92	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
93	#11761615	—	4	4	no	—
94	D₅₃₆	—	2	2	no	—
95	D₅₃₆	—	2	2	no	—
96	D₅₃₆	—	8	1	no	—
97	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
98	C₈×D₆₇	—	2	2	no	—
99	C₈×D₆₇	—	2	2	no	—
100	C₄⋊Dic₆₇	—	2	2	no	—
101	C₄⋊Dic₆₇	—	4	1	no	—
102	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
103	Dic₆₇⋊C₄	—	4	2	no	—
104	(C₂×D₆₇)⋊C₄	—	4	2	no	—
105	(C₂×D₆₇)⋊C₄	—	4	2	no	—
106	(C₂×D₆₇)⋊C₄	—	4	2	no	—
107	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
108	C₄×Dic₆₇	—	4	2	no	—
109	C₆₇⋊C₁₆	—	2	2	no	—
110	C₂×Q₈×C₆₇	—	2	1	no	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	Q₁₆×C₆₇	—	2	1	no	—
113	Q₁₆×C₆₇	—	2	1	no	—
114	Q₁₆×C₆₇	—	2	1	no	—
115	Q₁₆×C₆₇	#41	1	2	yes	—
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	4	1	no	—
121	C₄⋊C₄×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	—	4	1	no	—
123	C₄²×C₆₇	—	4	1	no	—
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₁₆×C₆₇	—	2	1	no	—
126	C₁₆×C₆₇	—	2	1	no	—
127	C₂²×D₆₇	—	4	2	no	—
128	C₂²×D₆₇	C₂×D₈	1	2	yes	—
129	Q₈×C₆₇	—	2	2	no	—
130	Q₈×C₆₇	—	4	1	no	—
131	Q₈×C₆₇	—	4	1	no	—
132	Q₈×C₆₇	—	4	1	no	—
133	C₂×C₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₆₇	—	4	2	no	—
141	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
142	D₂₆₈	—	2	2	no	—
143	D₂₆₈	—	4	2	no	—
144	D₂₆₈	—	4	2	no	—
145	D₂₆₈	—	8	4	no	—
146	D₂₆₈	—	8	2	no	—
147	D₂₆₈	—	16	1	no	—
148	D₂₆₈	—	16	2	no	—
149	C₄×D₆₇	—	2	2	no	—
150	C₄×D₆₇	—	2	2	no	—
151	C₄×D₆₇	—	8	4	no	—
152	Dic₁₃₄	—	8	2	no	—
153	C₈×C₆₇	—	2	1	no	—
154	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₆₇⋊C₈	—	2	2	no	—
157	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
158	C₂×D₆₇	—	2	2	no	—
159	C₂×D₆₇	—	2	2	no	—
160	C₂×D₆₇	—	2	2	no	—
161	C₂×D₆₇	—	8	2	no	—
162	C₂×D₆₇	—	8	2	no	—
163	C₂×D₆₇	—	8	2	no	—
164	C₄×C₆₇	—	2	1	no	—
165	C₄×C₆₇	—	4	2	no	—
166	C₄×C₆₇	—	8	1	no	—
167	C₄×C₆₇	—	8	1	no	—
168	C₄×C₆₇	—	8	1	no	—
169	C₄×C₆₇	#508	1	2	yes	—
170	C₄×C₆₇	#546	1	1	yes	characteristic
171	C₄×C₆₇	#577	1	1	yes	characteristic
172	Dic₆₇	—	2	2	no	—
173	Dic₆₇	—	8	2	no	—
174	#33871	—	67	1	no	maximal, Sylow
175	C₂×C₆₇	#3294	1	1	yes	characteristic
176	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
177	C₂×C₆₇	#3416	1	1	yes	characteristic
178	D₆₇	—	4	2	no	—
179	D₆₇	—	16	2	no	—
180	#3342	—	67	1	no	—
181	#3287	—	67	1	no	—
182	#2240	—	67	1	no	—
183	#2234	—	67	2	no	—
184	#2225	—	67	1	no	—
185	#2217	—	67	2	no	—
186	#2207	—	67	1	no	—
187	#2189	—	67	2	no	—
188	C₄×Q₃₂	D₆₇	1	1	yes	characteristic, complemented
189	D₄×C₁₆	—	67	1	no	—
190	#2138	—	67	2	no	—
191	C₆₇	#33871	1	1	yes	characteristic, complemented, Sylow
192	C₂×#135	—	67	2	no	—
193	#556	—	134	1	no	—
194	#550	—	134	1	no	—
195	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic
196	C₂×SD₃₂	—	67	2	no	—
197	C₂²×C₁₆	—	67	2	no	—
198	#500	—	67	1	no	—
199	#493	—	67	1	no	—
200	#485	—	134	2	no	—
201	#475	—	134	1	no	—
202	#467	—	67	2	no	—
203	#466	—	67	2	no	—
204	#460	—	134	1	no	—
205	#449	—	134	2	no	—
206	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
207	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
208	C₈×D₄	—	67	1	no	—
209	#427	—	67	2	no	—
210	#367	—	67	2	no	—
211	#366	C₂×D₆₇	1	1	yes	characteristic
212	C₄⋊C₁₆	—	67	1	no	—
213	#358	—	67	2	no	—
214	#358	C₂×D₆₇	1	1	yes	characteristic
215	#358	C₂×D₆₇	1	1	yes	characteristic
216	#357	—	67	2	no	—
217	#357	—	67	2	no	—
218	#357	—	67	2	no	—
219	C₂²⋊C₁₆	—	67	2	no	—
220	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
221	C₂×#41	—	134	2	no	—
222	#135	—	268	2	no	—
223	#135	—	134	4	no	—
224	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
225	C₂×Q₁₆	C₂²×D₆₇	1	1	yes	characteristic
226	C₂×SD₁₆	—	134	2	no	—
227	C₂×D₈	—	67	2	no	—
228	C₂×D₈	—	134	1	no	—
229	C₂²×C₈	—	67	2	no	—
230	C₄⋊Q₈	—	134	1	no	—
231	#127	—	268	1	no	—
232	#127	—	134	1	no	—
233	#126	—	268	2	no	—
234	#124	—	268	1	no	—
235	C₂²⋊Q₈	—	134	2	no	—
236	#121	—	268	2	no	—
237	#121	—	134	2	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	—	268	1	no	—
241	C₄×D₄	—	268	1	no	—
242	C₄×D₄	—	67	1	no	—
243	#117	—	134	2	no	—
244	Q₃₂	—	2	2	no	—
245	SD₃₂	—	268	2	no	—
246	C₂×C₁₆	—	268	2	no	—
247	C₂×C₁₆	—	67	2	no	—
248	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
249	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
250	#107	—	67	2	no	—
251	#107	—	134	1	no	—
252	#107	C₂²×D₆₇	1	1	yes	characteristic
253	#107	C₂²×D₆₇	1	1	yes	characteristic
254	#106	—	134	2	no	—
255	C₄⋊C₈	—	67	1	no	—
256	#103	—	2	1	no	—
257	#103	—	2	1	no	—
258	#103	—	134	2	no	—
259	#102	—	134	2	no	—
260	C₂²⋊C₈	—	67	2	no	—
261	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
262	#41	—	268	4	no	—
263	#41	—	536	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	268	1	no	—
267	C₂×D₄	—	268	2	no	—
268	C₂×D₄	—	268	2	no	—
269	C₂×D₄	—	268	1	no	—
270	C₂×D₄	—	268	1	no	—
271	C₂×D₄	—	67	1	no	—
272	C₂×D₄	—	134	2	no	—
273	C₂²×C₄	—	268	2	no	—
274	C₂²×C₄	—	67	2	no	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	#11761628	1	2	yes	—
279	SD₁₆	—	268	4	no	—
280	D₈	—	536	1	no	—
281	D₈	—	134	2	no	—
282	D₈	—	134	2	no	—
283	C₂×C₈	—	67	2	no	—
284	C₂×C₈	—	134	2	no	—
285	C₂×C₈	—	134	2	no	—
286	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
287	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	268	2	no	—
293	C₄⋊C₄	—	268	1	no	—
294	C₄⋊C₄	—	67	1	no	—
295	C₄⋊C₄	—	134	2	no	—
296	C₂²⋊C₄	—	268	2	no	—
297	C₂²⋊C₄	—	268	2	no	—
298	C₂²⋊C₄	—	268	2	no	—
299	C₂²⋊C₄	—	67	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	268	2	no	—
303	C₄²	D₄×D₆₇	1	1	yes	characteristic
304	C₁₆	—	2	1	no	—
305	C₁₆	—	2	1	no	—
306	C₁₆	—	134	2	no	—
307	C₂³	—	268	2	no	—
308	C₂³	—	67	2	no	—
309	Q₈	—	2	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	536	2	no	—
314	D₄	—	268	2	no	—
315	D₄	—	268	2	no	—
316	D₄	—	536	4	no	—
317	D₄	—	536	2	no	—
318	D₄	—	1072	1	no	—
319	D₄	—	1072	2	no	—
320	D₄	—	134	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	268	2	no	—
326	C₂×C₄	—	536	4	no	—
327	C₂×C₄	—	67	2	no	—
328	C₂×C₄	—	134	2	no	—
329	C₂×C₄	—	134	2	no	—
330	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
331	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	134	2	no	—
335	C₈	C₂×#11761628	1	1	yes	characteristic
336	C₈	#14524089	1	1	yes	characteristic
337	C₂²	—	536	2	no	—
338	C₂²	—	536	2	no	—
339	C₂²	—	536	2	no	—
340	C₂²	—	134	2	no	—
341	C₂²	—	134	2	no	—
342	C₂²	—	134	2	no	—
343	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₄	—	536	2	no	—
350	C₄	—	134	2	no	—
351	C₄	#16863829	1	2	yes	—
352	C₄	#16864042	1	1	yes	characteristic
353	C₄	#16864203	1	1	yes	characteristic
354	C₂	—	268	2	no	—
355	C₂	—	1072	2	no	—
356	C₂	#22572024	1	1	yes	characteristic
357	C₂	C₂×#16863829	1	1	yes	characteristic
358	C₂	#22573352	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×Q₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	#22571968	C₂
hypercenter	C₄×Q₃₂	D₆₇
hypocenter	C₆₇	#33871
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22572641	C₂	semidirect
3. #22562433	C₂	semidirect
4. #22562403	C₂	semidirect
5. #22562351	C₂	semidirect
6. #22562308	C₂	semidirect
7. #22562249	C₂	semidirect
8. #22562148	C₂	semidirect
9. C₁₆×D₂₆₈	C₂	semidirect
10. #22561915	C₂	semidirect
11. C₄×Q₃₂×C₆₇	C₂	semidirect
188. C₄×Q₃₂	D₆₇	semidirect
191. C₆₇	#33871	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22572641	2
#22571968	2
#22562433	2
#22562403	2
#22562351	2
#22562308	2
#22562249	2
#22562148	2
C₁₆×D₂₆₈	2
#22561915	2
C₄×Q₃₂×C₆₇	2
#33871	67